Exponential smoothing
see pdf notes for details on additive and multiplicative models.
ETS(error, trend, season)
Each error, trend and season can be additive/multiplicative/none, or damped.
report(fit)
Series: Pop
Model: ETS(A,A,N)
Smoothing parameters:
alpha = 0.9999
beta = 0.3266366
Initial states:
l b
10.05414 0.2224818
sigma^2: 0.0041
AIC AICc BIC
-76.98569 -75.83184 -66.68347
ETS(A,A,N) : A stands for “additive”, M for “multiplicative”, and N for “none”. Damped is “Ad” or “Md” for additive and multiplicative respectively.
\(\sigma^2\) is the variance of the residuals


Default chooses best (which is an AAN) in this case.
aus_economy %>%
model(
mod = ETS(Pop)
) %>%
report()
Series: Pop
Model: ETS(A,A,N)
Smoothing parameters:
alpha = 0.9999
beta = 0.3266366
Initial states:
l b
10.05414 0.2224818
sigma^2: 0.0041
AIC AICc BIC
-76.98569 -75.83184 -66.68347
A damped example: it’s leveling off.

An example with multiple series (and multiple models, best are selected automatically).
Lab session 14
Try foreasting Chinese GDP from global_economy using an ETS model.

Example: Australian holiday tourism. Observe all the unique combinations of models generated.
fit %>% pull(mod)
[1] ETS(A,N,A) ETS(A,A,N) ETS(M,N,A) ETS(M,N,A) ETS(M,N,M) ETS(A,N,A) ETS(M,N,M)
[8] ETS(M,N,A) ETS(A,N,A) ETS(A,N,N) ETS(M,N,N) ETS(M,N,M) ETS(A,A,N) ETS(A,N,A)
[15] ETS(M,N,A) ETS(A,N,N) ETS(M,N,M) ETS(M,N,A) ETS(M,N,M) ETS(M,N,M) ETS(M,N,A)
[22] ETS(M,N,N) ETS(M,N,A) ETS(M,N,M) ETS(A,N,A) ETS(M,N,A) ETS(M,N,A) ETS(M,N,A)
[29] ETS(M,N,A) ETS(A,N,A) ETS(M,N,A) ETS(M,N,A) ETS(A,N,A) ETS(A,N,N) ETS(A,N,A)
[36] ETS(M,N,M) ETS(M,N,A) ETS(M,N,M) ETS(M,A,A) ETS(M,A,A) ETS(M,N,A) ETS(M,N,A)
[43] ETS(M,N,A) ETS(A,N,A) ETS(M,N,A) ETS(A,N,A) ETS(A,N,N) ETS(M,N,A) ETS(A,N,A)
[50] ETS(M,A,A) ETS(M,N,M) ETS(A,N,N) ETS(M,N,M) ETS(M,N,M) ETS(A,N,A) ETS(M,N,A)
[57] ETS(A,N,A) ETS(A,N,A) ETS(M,N,A) ETS(M,N,M) ETS(M,N,A) ETS(M,N,M) ETS(A,N,A)
[64] ETS(M,N,A) ETS(M,N,M) ETS(M,N,N) ETS(M,N,A) ETS(M,N,A) ETS(M,N,A) ETS(A,A,A)
[71] ETS(M,N,A) ETS(M,A,A) ETS(M,N,M) ETS(M,N,M) ETS(A,A,N) ETS(M,N,A)
“Best” method chosen by AICc. See slide 39 of lecture 7. Paper for the method of “automatic forecasting” using AICc.
ETS won’t work well for data less than monthly level.
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